6.1 Polynomial Functions

Size: px

Start display at page:

Download "6.1 Polynomial Functions"

Thomas Ball
5 years ago
Views:

1 6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and p n 0. The degree of the polynomial is n. The coefficient p 0 is called the constant term, and the p n is called the leading coefficient. Behavior of Polynomial Functions Definition. For a function f(x), the phrase as x, f(x) means that as x goes to infinity (becomes arbitrarily large), the function values f(x) go to infinity (become arbitrarily large). End Behavior for Polynomial p(x) with highest degree term p n x n : Note. The end behavior for p(x) = p n x n + p n 1 x n p 2 x 2 + p 1 x + p 0 is determined by p n x n only. All we need to know is if p n is positive or negative and whether n is odd or even. Example 2. Describe the end behavior of the polynomials below. (a) p(x) = 5x 7x 37 11x 41 12x 111 1

2 (b) p(x) = (3x 5 4x)(8x 17 5x 6 ) (c) p(x) = ( 3x 7 5x)(8x + 10x 11 ) Example 3. If p(x) is a polynomial of degree 113 and the values of p(x) go to as x goes to, then what happens to p(x) for large negative values of x? Example 4. If p(x) is a polynomial of degree 24 and the values of p(x) go to as x goes to, then what happens to p(x) for large positive values of x? 2

3 6.2 Rational Functions Definition. A rational function is the quotient of two polynomials. That is, if r(x) is a rational function, then there are two polynomials p(x) and such that r(x) = p(x) Basic Facts: (1) The domain of a rational function is the set of real numbers x such that the denominator is not 0, i.e., = 0. (2) The x-intercepts of a rational function r(x) are the real numbers x, where the numerator is 0, and the denominator is not zero, i.e., p(x) = 0 and 0. Recall, the x-intercepts are also called zeros. (3) The y-intercept is the y-value when x = 0, i.e., the y-intercept is r(0). Note there is at most one y-intercept because r(x) is a function and must pass the vertical line test. (If 0 is not in the domain of r(x), there is no y-intercept.) 1. Find the domain, x-intercepts, and y-intercept for the function f(x) = 9(x + 3)(2x 7)(x + 1). x(4x 1)(x + 3) End behavior of rational functions Generalized Technique for the end behavior of a rational function: Find the term with highest power in the numerator; Find the term with highest power in the denominator; Check the end behavior of their quotient. 2. Determine the end behavior of the following functions: (a) f(x) = x2 7x x 5 + 4x 3 x 2 (b) f(x) = 8 5x2 7x 3 4x 10 5x x (c) f(x) = 5x30 x x 25 3

4 Horizontal Asymptotes Definition. If as x, f(x) a or as x, f(x) a, then the line y = a is a horizontal asymptote. Note. A rational function has at most one horizontal asymptote. Functions that are not rational may have two different horizontal asymptotes, one as x and one as x. A function CAN cross its horizontal asymptote! Horizontal Asymptotes: Let r(x) = p(x) be a rational function of x, where p(x) = p m x m + p m 1 x m p 0, = q n x n + q n 1 x n q 0 with m the degree of p(x) and n the degree of. Then we have three cases: 1. If m < n (i.e., the degree of the denominator is larger than the numerator), then the horizontal asymptote is y = If m = n (i.e., the degree of the denominator is the same as the numerator), then the horizontal asymptote is y = pm q n. 3. If m > n (i.e., the degree of the denominator is smaller than the numerator), then there is no horizontal asymptote. 3. Find the horizontal asymptotes of the following functions: (a) f(x) = x3 x 11 + x 2 5x 5 3x 2 7 (b) f(x) = 33x95 5x x 43 11x 111. (c) f(x) = 3x5 4x x 3 5x 5 + 7x + 15 (d) f(x) = 1 x n, where n is a positive integer. 4

5 4. Graph the function f(x) = 1 x+2 Vertical Asymptotes, and find its vertical asymptote. Definition. We say that a function y = f(x) has a vertical asymptote at x = a if the numbers f(x) become arbitrarily large as x approaches the value a from the right or the left. In other words, as x a, f(x) ±. Finding Vertical Asymptotes of r(x) = p(x) : 1. First cancel any common factors out of p(x). 2. Once p(x) is in lowest terms, this rational function will have a vertical asymptote at all values of x for which = 0. Therefore, set = 0, and solve for x. Note. (1) A rational function can have many different vertical asymptotes, but it only has one horizontal asymptote. (2) A function CANNOT cross its vertical asymptote (its undefined at this x-value). 5. Find the vertical asymptotes for the following functions (a). f(x) = 3x3 + 17x 2 28x 2x 3 19x 2 + 9x (b). f(x) = x 1 x 2 x 5

6 Holes in the Graph What happens to the factors we can cancel out from the numerator and denominator? example, what does the graph f(x) = x 1 x 2 look like? x For Finding Holes of r(x) = p(x) : 1. First cancel any common factors out of p(x). 2. The rational function will have a hole at a if the original denominator was zero when x = a, but the simplified denominator is no longer zero. 3. Plug x = a into the simplified fraction to find the y-value of the hole. Steps. For rational functions r(x) = p(x), if q(a) = 0 then r(x) has either a hole or a vertical asymptote when x = a. Simplify the function r(x) and 1. If the denominator is still zero when x = a, then its a vertical asymptote. 2. If the denominator is no longer zero when x = a, then its a hole. 6. Does the function f(x) = 5(x 1)(x 3)2 (x + 4) have any holes? If so, where? 3(x 1)(x 3)(x + 4) 2 6

7 7. For the following rational functions, find the domain, x-intercepts, y-intercepts, holes, vertical asymptotes, horizontal asymptotes, and end behavior. (a). f(x) = 5(x 1)(x 3)2 (x + 4) 3(x 1)(x 3)(x + 4) 2 Domain: x-intercept(s): y-intercept: Holes: Vertical Asymptotes: Horizontal Asymptotes: as x, f(x) as x, f(x) (b). f(x) = 3x(2x 5)(x 3) (x 7)(3x 4)(x 3) Domain: x-intercept(s): y-intercept: Holes: Vertical Asymptotes: Horizontal Asymptotes: as x, f(x) as x, f(x) 7

Rational Functions 4.5

Math 4 Pre-Calculus Name Date Rational Function Rational Functions 4.5 g ( ) A function is a rational function if f ( ), where g ( ) and ( ) h ( ) h are polynomials. Vertical asymptotes occur at -values